# mapping class group of the Klein bottle

the Klein bottle is gotten from a quotient in the two-torus $S^1\times S^1$ via

$K=\frac{S^1\times S^1}{(z,w)\sim(-z,\bar{w})}$

so to get all the isotopy classes $[f]:K\to K$ one need to consider the auto-homeomorphisms $\phi:T\to T$ of the torus.

It is well known that each of those are determined by a two-by-two matrix of $GL_2(\mathbb{Z})$, i.e $(z,w)\to (z^aw^b,z^cw^d)$ where $\det\left(\begin{array}{cc}a&b\\c&d\end{array}\right)=ad-bc=\pm1$, so in seeking those matrices which obey the above gluing conditions we are compeled to analyse

$(-z)^a\bar{w}^b=-z^aw^b$

$(-z)^c\bar{w}^d=\bar{z}^c\bar{w}^d$

for each couple $z,w\in S^1$.

Then if we set $w=1$ this implies $(-1)^az^a=-z^a$ and $(-1)^a=-1$, so $a$ is odd. Also,  if $z=1$ then $\bar{w}^b=w^b$ and hence $b=0$ .

But $ad=\pm1$, then $\left(\begin{array}{cc}\pm1&0\\ 0&\pm1\end{array}\right)$ are the only matrices which preserve the gluing conditions.

It is easy to see that these four matrices form the famous 4-Klein group. So ${\cal{MCG}}(K)=\mathbb{Z}_2\oplus\mathbb{Z}_2$.